A pyramid T.ABCD with base isosceles trapezoid. AB//DC, AB=9, CD=3, AD=BC=6, and TA=TB=TC=13. Find a. length of TD b. Height of pyramid?

Let P be the midpoint of AB and Q be mid point of CD

Let TH be the height of the pyramid

You can easily prove that PQ = 3√3

We know TP = √[13² - 1.5²] and TQ = √[13² - 4.5²], and

√[{13² - 1.5²} - h²] + √[{13² - 1.5²} - h²] = 3√3

Let L and M be midpoints of AD and BC.

It can be proved that LH = √12 and TL = √(13² -3²) = √160

TH² = TL² -LH² = 160 - 12 = 148 or TH = 2√37

Hint:

H is the centroid of the triangle ABC , and the height (of the pyramid) is TH , and angle ABC = 60 degrees